Consider the enumerative question of the number of lines in \(\mathbb {P}^2\)  going through a general point and intersecting \(D_1\)  and \(D_2\)  in one and two points respectively such that the cross ratio of the point mapping to the general point and the point of intersections with \(D_1\)  and \(D_2\)  satisfy a generic fixed cross-ratio. By the above theorem, together with Theorem \(9.3\)  of [1]}, this can be computed by computing the \(\vartheta _0\)  term of \(\vartheta _{v_1}\vartheta _{v_2}^2 \in R_{(X,D)}\) . To do this, we note the relation \(\vartheta _{v_2}\vartheta _{v_{1,i}+v_{2,i}} = t^{[l]} + \vartheta _{v_{1,i} + 2v_{2,i}}\) , which can be derived in a manner similar to Example \(1.21\)  of [1]}. Using this, observe:
\(tr(\vartheta _{v_2}^2\vartheta _{v_1}) = tr(\vartheta _{v_2}(\vartheta _{v_{1,1} + v_{2,1}} + \vartheta _{v_{1,2}+v_{2,2}})) = tr(\vartheta _{v_2}\vartheta _{v_{1,1} + v_{2,1}}) + tr(\vartheta _{v_2} \vartheta _{v_{1,2}+v_{2,2}}) = 2t^{[l]}.\) 
